Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World (54 page)

Opera geometrica
: Torricelli’s
Opera geometrica
can be found in volume 1 of Gino Loria and Giuseppe Vassura, eds.,
Opere di Evangelista Torricelli
(Faenza: G. Montanari, 1919–44). An Italian translation is in Lanfranco Belloni, ed.,
Opere scelte di Evangelista Torricelli
(Turin: Unione Tipografico-Editrice Torinese, 1975), pp. 53–483.

“De dimensione parabolae”
: For discussions of “De dimensione parabolae,” see François de Gandt, “Les indivisibles de Torricelli,” in de Gandt, ed.,
L’oeuvre de Torricelli
, pp. 152–53; and in de Gandt, “Naissance et métamorphose,” pp. 218–19.

that the ancients possessed a secret method
: Torricelli discusses this idea in the
Opera geometrica
, esp. in Loria and Vassura, eds.,
Opere
, vol. 1, pp. 139–40. The Italian translation is in Belloni, ed.,
Opere scelte
, p. 381.

“the Royal Road through the mathematical thicket”
: See Loria and Vassura, eds.,
Opere
, vol. 1, p. 173. Quoted in de Gandt, “Les indivisibles de Torricelli,” p. 153.

“We turn away from the immense ocean of Cavalieri’s Geometria”
: Loria and Vassura, eds.,
Opere
, vol. 1, p. 141.

Torricelli’s directness made his method far more intuitive
: See de Gandt, “Naissance et métamorphose,” p. 219.

three separate lists of paradoxes
: On Torricelli’s lists of paradoxes, see de Gandt, “Les indivisibles de Torricelli,” pp. 163–64.

the simplest one captures the essential problem
: Torricelli’s basic paradox is presented in a treatise entitled “De indivisibilium doctrina perperam usurpata,” in Loria and Vassura, eds.,
Opere
, vol. 1, part 2, p. 417.

“that indivisibles are all equal to each other”
: Torricelli’s discussion of unequal indivisibles can be found in Loria and Vassura, eds.,
Opere
, vol. 1, part 2, p. 320. It is quoted in de Gandt, “Les indivisibles de Torricelli,” p. 182.

“semi-gnomons”
: The diagrams here are derived from de Gandt, “Les indivisibles de Torricelli,” p. 187.

to calculate the slope of the tangent
: For a discussion of Torricelli’s method of tangents, see ibid., pp. 187–88, and idem, “Naissance et métamorphose,” pp. 226–29. De Gandt’s exposition is based on Torricelli’s
Opere
, ed. Loria and Vassura, vol. 1, part 2, pp. 322–33.

4. “Destroy or Be Destroyed”: The War on the Infinitely Small

“
one of the best books ever written on mathematics”
: Quoted in H. Bosmans, “André Tacquet (S. J.) et son traité d’arithmétique théorique et pratique,”
Isis
9 (1927): 66–82.

“either legitimate or geometrical”
: André Tacquet,
Cylindricorum et annularium libri IV
(Antwerp: Iacobum Mersium, 1651), pp. 23–24.

geometry formed the core of Jesuit mathematical practice
: On the persistence of the Euclidean tradition among Jesuit mathematicians through the eighteenth century, see Bosmans, “André Tacquet,” p. 77. Not coincidentally, some of the most popular textbooks on Euclidean geometry in that era were composed by Jesuits, including Honoré Fabri,
Synopsis geometrica
(Lyon: Antoine Molin, 1669); and Ignace-Gaston Pardies,
Elémens de géométrie
(Paris: Sébastien Maire-Cramoisy, 1671). Both textbooks were published repeatedly in the seventeenth and eighteenth centuries.

the struggle between geometry and indivisibles
: Tacquet,
Cylindricorum et annularium
, pp. 23–24.

the continuum is infinitely divisible
: Benito Pereira,
De communibus omnium rerum naturalium principiis
(Rome: Franciscus Zanettus, 1576). On Pereira’s discussion of the composition of the continuum, see Paolo Rossi, “I punti di Zenone,”
Nuncius
13, no. 2 (1998): 392–94.

“
great confusion and perturbation”
: Quoted in Feingold, “Jesuits: Savants,” p. 30.

The first decree by the Revisors General
: The Revisors’ condemnations of 1606 and 1608 can be found in the Jesuit archive ARSI (Archivum Romanum Societatis Iesu), manuscript FG656 A I, pp. 318–19.

Luca Valerio of the Sapienza University
: The book was Luca Valerio,
De centro gravitatis solidorum libri tres
(Rome: B. Bonfadini, 1604). On Valerio’s use of infinitesimal methods, see Carl B. Boyer,
History of the Calculus
(New York: Dover Publications, 1947), pp. 104ff.

experimenting with indivisibles
: On Galileo’s 1604 experimentation with indivisibles see Festa, “Quelques aspects de la controverse sur les indivisibles,” p. 196.

“the Archimedes of our age”
: Galileo,
Dialogues Concerning Two New Sciences
, p. 148.

“the continuum is composed of indivisibles”
: The first condemnation of 1615, dated April 4, is in ARSI manuscript FG 656A II, p. 456. The second, dated November 19, is in manuscript FG 656A II, p. 462.

Valerio had misread the signs
: On Valerio’s rise and fall, see David Freedberg,
The Eye of the Lynx: Galileo, His Friends, and the Beginnings of Modern Natural History
, (Chicago: University of Chicago Press, 2002), esp. pp. 132–34, as well as the online MacTutor biography of Valerio by J. J. O’Connor and E. F. Robertson at
http://www.gap-system.org/~history/Biographies/Valerio.html
. On his use of infinitesimals, see Boyer,
History of the Calculus
, pp. 104–106.

By relying on their hierarchical order
: On tensions between Jesuit intellectuals and the hierarchy’s efforts to control their scholarly work, see Feingold, “Jesuits: Savants”; and Marcus Hellyer, “‘Because the Authority of My Superiors Commands’: Censorship, Physics, and the German Jesuits,”
Early Science and Medicine
1, no. 3 (1996): 319–54.

he settled for a curt permission by the Jesuit provincial of Flanders
: In contrast, when Tacquet published his
Cylindricorum et annularium
four years later, also in Flanders, his license stated that his work had been read and approved by three mathematicians of the Society. St. Vincent’s book carried no such endorsement.

St. Vincent’s experience typifies the Jesuit attitude toward indivisibles
: On Gregory St. Vincent’s troubles, see Feingold, “Jesuits: Savants,” pp. 20–21; Herman Van Looey, “A Chronology and Historical Analysis of the Mathematical Manuscripts of Gregorius a Sancto Vincentio (1584–1667),”
Historia Mathematica
11 (1984): 58; Bosmans, “André Tacquet,” pp. 67–68; also Paul B. Bockstaele, “Four Letters from Gregorius a S. Vincentio to Christopher Grienberger,”
Janus
56 (1969): 191–202.

For the Jesuits, the choice could hardly have been worse
: On the changing political and cultural climate in Rome surrounding the election of Urban VIII, see Pietro Redondi,
Galileo Heretic
(Princeton, NJ: Princeton University Press, 1987), esp. pp. 44–61 and 68–106.

“bring down the pride of the Jesuits”
: This is from a letter by the Lincean Dr. Johannes Faber to Galileo from February 15, 1620. Quoted in Redondi,
Galileo Heretic
, p. 43.

the new Pope was amused and full of admiration
: Cesarini to Galileo, October 28, 1623. Quoted in Redondi,
Galileo Heretic
, p. 49.

The plot misfired badly
: On the Santarelli affair, see Bangert,
A History of the Society of Jesus
, pp. 200–201; and Redondi,
Galileo Heretic
, pp. 104–105.

Galileo even believed that he had been given implicit permission
: See Redondi,
Galileo Heretic
, p. 50.

The young aristocrat was a rising star in Roman intellectual circles
: On Pallavicino’s dissertation defense, see ibid., pp. 200–202. Father Grassi had been Galileo’s opponent in a dispute over the nature of comets and the chief target of
The Assayer
. His condemnation of the orthodoxy of Galileo’s atomism was contained in his 1626 book
Ratio ponderum librae et simbellae
, published under the pseudonym Lothario Sarsi.

Urban VIII had run out of options
: On the political crisis in Rome in 1631, and on Cardinal Borgia’s attack on Urban VIII, see Redondi,
Galileo Heretic
, pp. 229–31.

Galileo’s friends were running for cover
: Not all made it safely. In April of 1632, Giovanni Ciampoli, the most prominent Lincean in the Curia and personal secretary to the Pope himself, was given the impressive-sounding title of “Governor of Montalti di Castro” and exiled from Rome to the Apennines. He would never return.

Father Rodrigo de Arriaga of Prague
: On Rodrigo Arriaga, his
Cursus philosophicus
, and his views on the infinitely small, see Rossi, “I punti di Zenone,” pp. 398–99; Hellyer, “‘Because the Authority of My Superiors Commands,’” p. 339; Feingold, “Jesuits: Savants,” p. 28; Redondi,
Galileo Heretic
, pp. 241–42; and John L. Heilbron,
Electricity in the 17th and 18th Centuries
(Berkeley: University of California Press, 1979), p. 107.

Quite possibly he was influenced by his friend Gregory St. Vincent
: On Arriaga’s friendship with St. Vincent, see Van Looey, “A Chronology and Historical Analysis of the Mathematical Manuscripts of Gregorius a Sancto Vincentio,” p. 59.

“The permanent continuum can be constituted”
: The Revisors’ decree is preserved as manuscript FG 657, p. 183, in ARSI (Archivum Romanum Societatis Iesu), the archive of the Society of Jesus in Rome. It is also reproduced in Egidio Festa, “La querelle de l’atomisme,”
La Recherche
224 (September 1990): 1040; and quoted in French in Egidio Festa, “Quelques aspects de la controverse sur les indivisibles,” in M. Bucciantini and M. Torrini, eds.,
Geometria e atomismo nella scuola Galileana
(Florence: Leo S. Olschki, 1992), p. 198. Special thanks to Professor Carla Rita Palmerino of Radboud University Nijmegen, in the Netherlands, for making available to me her notes from the Jesuit archives.

he found himself writing to Father Ignace Cappon
: General Mutio Vitelleschi to Ignace Cappon, 1633, quoted in Michael John Gorman, “A Matter of Faith? Christoph Scheiner, Jesuit Censorship, and the Trial of Galileo,”
Perspectives on Science
4, no. 3 (1996): pp. 297–98. Also quoted in Feingold, “Jesuits: Savants,” p. 29.

On February 3, 1640
: ARSI manuscript FG 657, p. 481.

in January 1641:
Ibid., p. 381. Cited and discussed in Festa, “Quelques aspects,” pp. 201–202.

On May 12, 1643
: ARSI manuscript FG 657, p. 395.

In 1649
: Ibid., p. 475.

Arriaga’s views on the continuum were unequivocally condemned
: On Arriaga and the publishing history of his
Cursus philosophicus
, see Hellyer, “‘Because the Authority of My Superiors Commands,’” pp. 339–41.

Pallavicino was no ordinary novice
: On Pietro Sforza Pallavicino and his career, see Redondi,
Galileo Heretic
, pp. 264–65; Hellyer, “‘Because the Authority of My Superiors Commands,’” p. 339; Festa, “La querelle de l’atomisme,” pp. 1045–46; Festa, “Quelques aspects,” pp. 202–203; and Feingold, “Jesuits: Savants,” p. 29.

the marchese still considered himself a progressive thinker
: Redondi,
Galileo Heretic
, p. 265.

Pallavicino frequently came under the Revisors’ scrutiny
: On Pallavicino’s conflicts with the Revisors and General Carafa, see Claudio Costantini,
Baliani e i Gesuiti
(Florence: Giunti, 1969), esp. pp. 98–101.

Pallavicino forged ahead, lecturing on his unorthodox views
: Pallavicino hints at his troubles in Pietro Sforza Pallavicino,
Vindicationes Societatis Iesu
(Rome: Dominic Manephi, 1649), p. 225. Quoted and discussed in Festa, “Quelques aspects,” pp. 202–203.

“there are some in the Society who follow Zeno”
: Superior General Vincenzo Carafa to Nithard Biberus, March 3, 1649. In G. M. Pachtler, SJ, ed.,
Ratio studiorum et institutiones scholasticae Societatis Jesu
(Osnabrück: Biblio-Verlag, 1968), 3:76, doc. no. 41.

Ordinatio pro studiis superioribus
: For the text of the
Ordinatio
, see G. M. Pachtler, SJ, ed.,
Ratio studiorum
, vol. 3 (Berlin: Hofman and Comp., 1890), pp. 77–98. The sixty-five banned “philosophical” propositions are on pages 90–94, and an additional list of twenty-five banned “theological” propositions is on pages 94–96. For a discussion of the
Ordinatio
, its origins, and its effects, see Hellyer, “‘Because the Authority of My Superiors Commands,’” pp. 328–29. It is also mentioned in Feingold, “Jesuits: Savants,” p. 29; and Carla Rita Palmerino, “Two Jesuit Responses to Galileo’s Science of Motion: Honoré Fabri and Pierre le Cazre,” in M. Feingold, ed.,
The New Science and Jesuit Science: Seventeenth-Century Perspectives
(Dordrecht: Kluwer Academic Publishers, 2003), p. 187.

“The succession continuum”
: The propositions are listed in Pachtler, ed.,
Ratio studiorum
, p. 92.

5. The Battle of the Mathematicians

“the three Jesuits, Guldin, Bettini, and Tacquet”
: Stefano degli Angeli,
De infinitis parabolis
(Venice: Ioannem La Nou, 1659), under “Lectori Benevolo.”

It was also crucial to prove them mathematically wrong
: On Guldin, Bettini, and Tacquet as the Society’s agents sent to combat the method of indivisibles, see Redondi,
Galileo Heretic
, p. 291.